{"id":1510,"date":"2026-05-17T08:13:27","date_gmt":"2026-05-17T08:13:27","guid":{"rendered":"https:\/\/mymockmate.com\/notes\/?p=1510"},"modified":"2026-05-19T07:10:59","modified_gmt":"2026-05-19T07:10:59","slug":"ncert-class-10-maths-real-number-exercise-1-2-solutions-mymockmate","status":"publish","type":"post","link":"https:\/\/mymockmate.com\/notes\/ncert-class-10-maths-real-number-exercise-1-2-solutions-mymockmate\/","title":{"rendered":"NCERT Class 10 Maths Real Number Exercise 1.2 Solutions | MyMockMate"},"content":{"rendered":"\n
\n\n\n\n

Short Intro<\/h4>\n\n\n\n

In this post, students can find complete step-by-step solutions for Class 10 Maths Chapter 1 Real Numbers Exercise 1.2. All questions are solved using simple explanations based on the latest NCERT syllabus and CBSE pattern to help students prepare effectively for board exams.<\/p>\n\n\n\n

Exercise 1.2 focuses on irrational numbers and proofs using contradiction method.<\/p>\n\n\n\n

\n\n\n\n

Quick Information Box<\/h4>\n\n\n\n
Item<\/th>Details<\/th><\/tr><\/thead>
Board<\/td>CBSE<\/td><\/tr>
Class<\/td>10<\/td><\/tr>
Subject<\/td>Maths<\/td><\/tr>
Chapter<\/td>Real Numbers<\/td><\/tr>
Exercise<\/td>1.2<\/td><\/tr>
Topic<\/td>Irrational Numbers<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
\n\n\n\n

Concepts Used (Topics Covered)<\/h4>\n\n\n\n
    \n
  • Irrational Numbers<\/li>\n\n\n\n
  • Proof by Contradiction<\/li>\n\n\n\n
  • Rational and Irrational Numbers<\/li>\n\n\n\n
  • Properties of Square Roots<\/li>\n\n\n\n
  • Fundamental Theorem of Arithmetic<\/li>\n<\/ul>\n\n\n\n

    The exercise is based on irrationality proofs discussed in Chapter 1.<\/p>\n\n\n\n

    \n\n\n\n

    Important Formulas<\/h4>\n\n\n\n

    Rational Number Form<\/h6>\n\n\n\n

    p<\/mi>q<\/mi><\/mfrac>,<\/mo> <\/mtext>q<\/mi>\u2260<\/mo>0<\/mn><\/mrow>\\frac{p}{q},\\ q\\neq0<\/annotation><\/semantics><\/math><\/p>\n\n\n\n
    \n\n\n\n
    Irrational Number Concept<\/h6>\n\n\n\n

    p<\/mi><\/msqrt> is irrational for prime <\/mtext>p<\/mi><\/mrow>\\sqrt{p}\\ \\text{is irrational for prime}\\ p<\/annotation><\/semantics><\/math><\/p>\n\n\n\n
    \n\n\n\n

    Questions & Step-by-step Solutions<\/h4>\n\n\n\n

    Question 1<\/h4>\n\n\n\n

    Prove that \u221a5 is irrational.<\/h6>\n\n\n\n
    Solution<\/h6>\n\n\n\n

    Assume, to the contrary, that:<\/p>\n\n\n\n

    5<\/mn><\/msqrt>=<\/mo>a<\/mi>b<\/mi><\/mfrac><\/mrow>\\sqrt{5}=\\frac{a}{b}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

    where:<\/p>\n\n\n\n

      \n
    • a and b are coprime integers<\/li>\n\n\n\n
    • b \u2260 0<\/li>\n<\/ul>\n\n\n\n

      Squaring both sides:<\/p>\n\n\n\n

      5 = a\u00b2\/b\u00b2\n<\/code><\/pre>\n\n\n\n
      a\u00b2 = 5b\u00b2\n<\/code><\/pre>\n\n\n\n
      Thus, 5 divides a\u00b2.<\/p>\n\n\n\n
      By the theorem:<\/p>\n\n\n\n
      If p divides a\u00b2, then p divides a.\n<\/code><\/pre>\n\n\n\n
      Therefore, 5 divides a.<\/p>\n\n\n\n
      Let:<\/p>\n\n\n\n
      a = 5c\n<\/code><\/pre>\n\n\n\n
      Substituting:<\/p>\n\n\n\n
      25c\u00b2 = 5b\u00b2\n<\/code><\/pre>\n\n\n\n
      5c\u00b2 = b\u00b2\n<\/code><\/pre>\n\n\n\n
      Thus, 5 divides b\u00b2 and hence 5 divides b.<\/p>\n\n\n\n
      Therefore, a and b have a common factor 5.<\/p>\n\n\n\n
      But this contradicts the assumption that a and b are coprime.<\/p>\n\n\n\n
      Hence,<\/p>\n\n\n\n
      Final Answer:<\/p>\n\n\n\n
      \u221a5 is irrational.\n<\/code><\/pre>\n\n\n\n
      \n\n\n\n
      Question 2<\/h4>\n\n\n\n
      Prove that 3 + 2\u221a5 is irrational.<\/h6>\n\n\n\n
      Solution<\/h6>\n\n\n\n
      Assume, to the contrary, that:<\/p>\n\n\n\n
      3 + 2\u221a5 is rational\n<\/code><\/pre>\n\n\n\n
      Subtracting 3 from both sides:<\/p>\n\n\n\n
      2\u221a5 is rational\n<\/code><\/pre>\n\n\n\n
      Dividing by 2:<\/p>\n\n\n\n
      \\sqrt{5}=\\frac{\\text{rational}}{2}<\/p>\n\n\n\n
      This implies \u221a5 is rational.<\/p>\n\n\n\n
      But from Question 1, we know:<\/p>\n\n\n\n
      \u221a5 is irrational.\n<\/code><\/pre>\n\n\n\n
      This contradiction arises because our assumption was wrong.<\/p>\n\n\n\n
      Hence,<\/p>\n\n\n\n
      Final Answer:<\/p>\n\n\n\n
      3 + 2\u221a5 is irrational.\n<\/code><\/pre>\n\n\n\n
      \n\n\n\n
      Question 3<\/h4>\n\n\n\n
      Prove that the following are irrational numbers.<\/h6>\n\n\n\n
      \n\n\n\n
      (i) 1\/\u221a2<\/h6>\n\n\n\n
      Solution<\/h6>\n\n\n\n
      Assume:<\/p>\n\n\n\n
      \\frac{1}{\\sqrt{2}}<\/p>\n\n\n\n
      is rational.<\/p>\n\n\n\n
      Then:<\/p>\n\n\n\n
      1\/\u221a2 = a\/b\n<\/code><\/pre>\n\n\n\n
      Rearranging:<\/p>\n\n\n\n
      \u221a2 = b\/a\n<\/code><\/pre>\n\n\n\n
      Thus \u221a2 becomes rational.<\/p>\n\n\n\n
      But \u221a2 is irrational.<\/p>\n\n\n\n
      Contradiction.<\/p>\n\n\n\n
      Hence,<\/p>\n\n\n\n
      1\/\u221a2 is irrational.\n<\/code><\/pre>\n\n\n\n
      \n\n\n\n
      (ii) 7\u221a5<\/h4>\n\n\n\n
      Solution<\/h6>\n\n\n\n
      Assume:<\/p>\n\n\n\n
      7\u221a5 is rational.\n<\/code><\/pre>\n\n\n\n
      Dividing by 7:<\/p>\n\n\n\n
      \u221a5 becomes rational.\n<\/code><\/pre>\n\n\n\n
      But \u221a5 is irrational.<\/p>\n\n\n\n
      Contradiction.<\/p>\n\n\n\n
      Hence,<\/p>\n\n\n\n
      7\u221a5 is irrational.\n<\/code><\/pre>\n\n\n\n
      \n\n\n\n
      (iii) 6 + \u221a2<\/h4>\n\n\n\n
      Solution<\/h6>\n\n\n\n
      Assume:<\/p>\n\n\n\n
      6 + \u221a2 is rational.\n<\/code><\/pre>\n\n\n\n
      Subtracting 6:<\/p>\n\n\n\n
      \u221a2 becomes rational.\n<\/code><\/pre>\n\n\n\n
      But \u221a2 is irrational.<\/p>\n\n\n\n
      Contradiction.<\/p>\n\n\n\n
      Hence,<\/p>\n\n\n\n
      6 + \u221a2 is irrational.\n<\/code><\/pre>\n\n\n\n
      \n\n\n\n
      Common Mistakes<\/h4>\n\n\n\n
        \n
      • Forgetting the contradiction step<\/li>\n\n\n\n
      • Not assuming numbers are coprime<\/li>\n\n\n\n
      • Incorrect rearrangement of equations<\/li>\n\n\n\n
      • Confusing rational and irrational numbers<\/li>\n<\/ul>\n\n\n\n
        \n\n\n\n
        Exam Tips<\/h4>\n\n\n\n
          \n
        • Always start proofs with \u201cAssume, to the contrary\u2026\u201d<\/li>\n\n\n\n
        • Write contradiction clearly<\/li>\n\n\n\n
        • Mention theorem properly<\/li>\n\n\n\n
        • Practice square root irrationality proofs regularly<\/li>\n<\/ul>\n\n\n\n
          \n\n\n\n
          Practice MCQs<\/h4>\n\n\n\n
          MCQ 1<\/h6>\n\n\n\n
          \u221a7 is:<\/p>\n\n\n\n
          A. Rational
          B. Irrational
          C. Integer
          D. Whole number<\/p>\n\n\n\n
          Answer:<\/p>\n\n\n\n
          B. Irrational\n<\/code><\/pre>\n\n\n\n
          \n\n\n\n
          MCQ 2<\/h6>\n\n\n\n
          Which of the following is irrational?<\/p>\n\n\n\n
          A. 3\/5
          B. 0.25
          C. \u221a11
          D. 7<\/p>\n\n\n\n
          Answer:<\/p>\n\n\n\n
          C. \u221a11\n<\/code><\/pre>\n\n\n\n
          \n\n\n\n
          MCQ 3<\/h6>\n\n\n\n
          The sum of a rational and irrational number is:<\/p>\n\n\n\n
          A. Rational
          B. Irrational
          C. Integer
          D. Natural<\/p>\n\n\n\n
          Answer:<\/p>\n\n\n\n
          B. Irrational\n<\/code><\/pre>\n\n\n\n
          \n\n\n\n
          MCQ 4<\/h6>\n\n\n\n
          Which theorem is used in irrationality proofs?<\/p>\n\n\n\n
          A. Pythagoras Theorem
          B. Fundamental Theorem of Arithmetic
          C. Euclid\u2019s Theorem
          D. Binomial Theorem<\/p>\n\n\n\n
          Answer:<\/p>\n\n\n\n
          B. Fundamental Theorem of Arithmetic\n<\/code><\/pre>\n\n\n\n
          \n\n\n\n
          11. FAQ Section<\/h4>\n\n\n\n
          What is an irrational number?<\/h6>\n\n\n\n
          A number that cannot be expressed in the form:<\/p>\n\n\n\n
          p\/q<\/p>\n\n\n\n
          where q \u2260 0.<\/p>\n\n\n\n
          \n\n\n\n
          Is \u221a5 rational?<\/h6>\n\n\n\n
          No, \u221a5 is irrational.<\/p>\n\n\n\n
          \n\n\n\n
          Which method is used in Exercise 1.2?<\/h4>\n\n\n\n
          Proof by contradiction method.<\/p>\n\n\n\n
          \n\n\n\n
          Is Exercise 1.2 important for board exams?<\/h4>\n\n\n\n
          Yes, irrationality proofs are frequently asked in CBSE board exams.<\/p>\n\n\n\n
          \n\n\n\n
          12. CTA (Call To Action)<\/h4>\n\n\n\n
          \ud83d\udcd8 Practice More Questions & Improve Your Preparation!<\/p>\n\n\n\n
          \u2705 Attempt Chapter-wise Mock Tests
          \u2705 Solve Important MCQs
          \u2705 Download Notes PDF
          \u2705 Get Instant Performance Analysis<\/p>\n\n\n\n
          Start practicing now on MyMockMate.<\/p>\n\n\n\n
          <\/p>\n\n    
          \n\n\t\t\n        
            \n\t\t\t        <\/ul>\n    <\/div> \n","protected":false},"excerpt":{"rendered":"
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